Topics Resisters in Series Total Series Resistance Current Ohm’s Law Voltage Sources Kirchhoff’s Voltage Law Voltage Dividers

All circuits have three common attributes. These are: 1. A source of voltage. 2. A load. 3. A complete path.

A series circuit is one that has only one current path. VS R2 R1 R2 R3 R2 VS VS R3 R3

Connect each group to form a series circuit

The total resistance of resistors in series is Series Additive Rule The total resistance of resistors in series is the sum of the individual resistors. RT = R1 + R2 + R3 + RN If all resistors have the same value: RT = nR

Series Additive Rule The resistors in a series circuit are 680 W, 1.5 kW, and 2.2 kW. What is the total resistance? 4.38 kW

Measuring Resistance in a Series Circuit 56 kΩ

Because there is only one path, the current everywhere is the same. Series circuit rule for current Because there is only one path, the current everywhere is the same. 2.0 mA For example, the reading on the first ammeter is 2.0 mA, What do the other meters read? 2.0 mA 2.0 mA 2.0 mA

Current entering any point in a series circuit is the same as the current leaving that point.

Ohm’s law for Series Circuits IT = VS = system voltage; RT = Total Resistance IT = VX = voltage drop across one resistor; RX = resistance of that resistor ∴ Vx = IT * RX

Calculate IT for the circuit below: 200 18.5 mA

Series Resistors as current limiting devices

Voltage sources in Series

Voltage sources in series add algebraically. VS(Total) = VR1 + VR2 +…+ VRN The total voltage of the sources shown is 9 V

Batteries connected in opposite directions have a total voltage equal the algebraic sum of the voltages

VS = V1 + V2 + V3 + … + V4 Kirchhoff’s voltage law Kirchhoff’s voltage law (KVL) is generally stated as: The sum of all the voltage drops around a single closed path in a circuit is equal to the total source voltage in that closed path. VS = V1 + V2 + V3 + … + V4

Kirchhoff’s voltage law (KVL) applies to all circuits Must apply it to only one closed path In a series circuit, this is the entire circuit. A mathematical shorthand way of writing KVL is VS - V1 - V2 - V3 - … - V4 = 0

Kirchhoff’s Voltage Law

A string of resistors in series divides the voltage Voltage divider A string of resistors in series divides the voltage

Voltage divider rule The voltage drop across any given resistor in a series circuit is equal to the ratio of that resistor to the total resistance, multiplied by source voltage. Assume R1 is twice the size of R2. What is the voltage across R1? 8 V

What is the voltage across R2? VRx = The total resistance is 25 kW. Applying the voltage divider formula: Notice that 40% of the source voltage is across R2, which represents 40% of the total resistance. 8.0 V

Calculate the value of R4 1. Calculate VR1, VR2 & VR3 VR1 =IT * R1 2. Solve for VR4 using KVL 3. Solve for R4 4. R4 = V4/IT = 430 ohms Alternate Method: IT = VS/(1570 + R4) 20

Calculate voltage drop between: a to b; b to c; c to d; a to c; b to d; a to d 50 13.2V 46V

The potentiometer used as a voltage divider.

Voltage dividers can be set up for a variable output using a potentiometer. In the circuit shown, the output voltage is variable. What is the largest output voltage available at VOUT? 5.0 V

Adjusting the voltage divider.

A variable voltage divider used for volume control in a radio receiver.

A potentiometer voltage divider used as a level sensor. As liquid level decreases, the VOUTPUT decreases

Application of the voltage divider concept to measure strain

POWER IN A SERIES CIRCUIT PTotal = PR1 + PR2 + PR3 +… + PRN PTotal =

Use the voltage divider rule to find V1 and V2 Use the voltage divider rule to find V1 and V2. Then find the power in R1 and R2 and PT. Applying the voltage divider rule: 11.75 V The power dissipated by each resistor is: 8.25 V 0.29 W } PT = 0.5 W 0.21 W

Ground Path in a DC Circuit

The ground point does not affect: 1. the current in the circuit or 2. the voltage drops across the resistors.

Current ceases when an open occurs.

The source voltage appears across the open series resistor.

Troubleshooting a series circuit for an open using half-splitting.

The effect of a short in a series circuit.

Troubleshooting a series circuit for a short using half-splitting.

Voltage measurements Voltage is relative and is measured with respect to another point in the circuit. Voltages that are given with respect to ground are shown with a single subscript. VA means the voltage at point A with respect to ground (called reference ground). VB means the voltage at point B with respect to ground. VAB means the voltage between points A and B. What are VA, VB, and VAB for the circuit shown? VA = 12 V VB = 8 V VAB = 4 V

Ground reference is not always at the lowest point in a circuit Ground reference is not always at the lowest point in a circuit. Assume the ground is moved to B as shown. What are VA, VB, and VC for the circuit? VA = 4 V VB = 0 V VC = -8 V Has VAB changed from the previous circuit? No, it is still 4 V

If R2 is open, what are VA, VB, and VC for the circuit? If R2 is open, there is no current. VB = 0 V because it is ground VA = 0 V because it has the same potential as VB. VC = -12 V because the source voltage is across the open.

REVIEW Tabulating current, resistance, voltage and power is a useful way to summarize parameters in a series circuit. Complete the parameters listed in the Table. I1= R1= 0.68 kW V1= P1= I2= R2= 1.50 kW V2= P2= I3= R3= 2.20 kW V3= P3= IT= RT= 4.38 kW VS= 12 V PT= 2.74 mA 1.86 V 5.1 mW 2.74 mA 4.11 V 11.3 mW 2.74 mA 6.03 V 16.5 mW 2.74 mA 32.9 mW

Key Terms Series Kirchhoff’s voltage law Voltage divider In an electric circuit, a relationship of components in which the components are connected such that they provide a single path between two points. A law stating that (1) the sum of the voltage drops around a closed loop equals the source voltage in that loop or (2) the algebraic sum of all of the voltages (drops and source) is zero. A circuit consisting of series resistors across which one or more output voltages are taken.

Key Terms Reference ground Open Short The metal chassis that houses the assembly or a large conductive area on a printed circuit board is used as a common or reference point; also called common. A circuit condition in which the current path is broken. A circuit condition in which there is zero or an abnormally low resistance between two points; usually an inadvertent condition.

Quiz 1. In a series circuit with more than one resistor, the current is a. larger in larger resistors b. smaller in larger resistors c. always the same in all resistors d. there is not enough information to say

Quiz 2. In a series circuit with more than one resistor, the voltage is a. larger across larger resistors b. smaller across larger resistors c. always the same across all resistors d. there is not enough information to say

Quiz 3. If three equal resistors are in series, the total resistance is a. one third the value of one resistor b. the same as one resistor c. three times the value of one resistor d. there is not enough information to say

Quiz 4. A series circuit cannot have a. more than two resistors b. more than one voltage source c. more than one path d. all of the above

Quiz 5. In a closed loop, the algebraic sum of all voltages (both sources and drops) a. is zero b. is equal to the smallest voltage in the loop c. is equal to the largest voltage in the loop d. depends on the source voltage

Quiz 6. The current in the 10 kW resistor is a. 0.5 mA b. 2.0 mA c. 2.4 mA d. 10 mA

Quiz 7. The output voltage from the voltage divider is a. 2.0 V b. 4.0 V c. 12 V d. 20 V

Quiz 8. The smallest output voltage available from the voltage divider is a. 0 V b. 1.5 V c. 5.0 V d. 7.5 V

Quiz 9. The total power dissipated in a series circuit is equal to the a. power in the largest resistor b. power in the smallest resistor c. average of the power in all resistors d. sum of the power in all resistors

Quiz 10. The meaning of the voltage VAB is the voltage at a. Point A with respect to ground b. Point B with respect to ground c. The average voltage between points A and B. d. The voltage difference between points A and B.

Quiz Answers: 1. c 2. a 3. c 4. c 5. a 6. b 7. b 8. a 9. d 10. d